Chapter 6 - Limiting factors Flashcards
What is the contribution per unit of limiting factor for product T (q137)
The reason why we need to find the contribution per limiting factor is we need to work out which product to produce for example in question 137 it is asking for the limiting factor of product t. if I look at the contribution For product T then I can see that it is £57 versus £37 for product b. So at this point I would assume that product T is the better option however it is 3 kg for a product T and 2 kilograms for product b And as materials is the limiting factor if I divide the contribution by the amount of materials then you’ll notice that product t Is £17 and product B is £18.50 and therefore product B is the better option.
What is linear programming
Unlike with key factor analysis and throughput accounting where there is only one limited resource, Linear programming looks at situations where there is more than one limited resource
What are the steps to follow in linear programming
Define the unknown in terms of symbols
S = number of standard chairs
E = number of executive chairs
C= total contribution
Formulate equations for the constraints
Material 2S + 4E = < 80
Labour 5S + 6E = < 180
Demand E < 10
Non negativity S > 0
E> 0
Formulate an equation for the objective
(Our objective is to maximise the contribution which is C)
C = 6S + 9E ( 6 and 9 are the contribution per unit)
Graph the constraints and the objective
Find the optimum solution
What is the feasible area/feasible region
This is the region on the graph which satisfies all constraints. (The bit in red that had ABCD)
What is the objective line
The linear programming graph has both constraint lines for example materials labour and demand and Has a feasible area/region where the combination must lie and it also has an objective line which is the contribution gradient. Remember that even though we don’t know the contribution total we know the contribution per unit so we can put in any number for the contribution total so that we can plots the line on the graph.
For example if C = 6S +9E then if c = 90 we could do 15s + 0e and 0s + 15e to get the gradient
How do we know where the objective line should be on the graph
As stated before we don’t know what the contribution total is but we do know that whatever contribution we put the gradients on the graph will stay the same. In the example before we use a contribution of 90 however if we use the contribution of 180 then the gradient would stay the same put the line would move up. Therefore we want the highest points on the graph because the higher the points the higher the contribution hence this is why the area B what is the highest points on example. Because we want to go as high as we can go without leaving the feasible area
Therefor point b is the optimum
How to work out slack
Slack or spare capacity Is the Amount we have left once we’ve decided on the optimum combination for the limited resources.
Two ways to work out slack would be firstly if there is a graph then is the optimum points on the graph sitting on the constraint line if so then this is the maximum for that constraint
Arithmetically let’s say for example if the maximum material is 80 kg and originally we worked out the constraints equation for materials as 2S + 4E and if now now the optimum contribution is 2 x 30 and 4 x 5 we know that we need to make 30 standard chairs which requires 2 kg of materials each and 5 executive chairs that require 4 kg each totalling 80kg so the slack is 0
What is shadow prices/dual prices
In real life there is unlikely to be any true limited resource it is almost always possible to get more but we are likely to pay premium for it for example the supplier labour may be limited by the length of the working week but we can get more hours if we are prepared to pay overtime.
The shadow price of a limited resource is the most extras that we would be prepared to pay for one extra units are the limited resource we calculate by calculating the extra profit that would result if we had one extra units of limited resource therefore the maximum we would be Prepared to pay would be the full additional profit amount. Otherwise if The premium we were paying for the extra unit of constraints was more than the profit made on the difference then we will be making less of a contribution and therefore profits than we would’ve done if we hadn’t of bought the extra constraint
If the slack was five for demand what would a shadow price be
The shadow price would be zero. Because we have not exhausted the constraint For example if Demand was 10 and we were only using 5 what is the point of paying extra for more than 10 ?
